High school geometry
Review the formulas for the volume of prisms, cylinders, pyramids, cones, and spheres.
It may seem at first like there are lots of volume formulas, but many of the formulas share a common structure.
Prisms and prism-like figures
We always measure the height of a prism perpendicularly to the plane of its base. That's true even when a prism is on it's side or when it tilts (an oblique prism).
Often, we first learn about volume using rectangular prisms (specifically right rectangular prisms), such as by building the prism out of cubes.
Note that any face of a rectangular prism could be its base, as long as we measure the height of the prism perpendicularly to that face.
A cube with a length of l, a width of w, and a height of h.
A triangular prism has a base shaped like a triangle.
A triangular prism with a triangular base of base b, a triangular base of height h, and the length of l.
A circular cylinder is a prism-like figure that has a base shaped like a circle.
A cylinder with a radius r and a height h.
In oblique prisms, the bases are in parallel planes,
We still calculate the volume in exactly the same way because of Cavalieri's principle.
Which expression gives the volume of the oblique rectangular prism?
A oblique rectangular prism with its rectangular base with a length of two units and a width of one point five units. The prism's slanted height is five units. Its vertical height is four units.
Pyramids and pyramid-like figures
We also measure the height of a pyramid perpendicularly to the plane of its base. Because of Cavalieri's principle, the same volume formula works for right and oblique pyramid-like figures.
A rectangle-based pyramid has a base shaped like a rectangle.
A rectangular pyramid with the rectangular base's length of l units and width of w units. It has a vertical height of h units.
A circular cone is a pyramid-like figure that has a base shaped like a circle.
A cone with a radius of r and a vertical height of h.
A sphere with a radius of r.
Want to join the conversation?
- Why is the Volume of sphere is divided by 3?(11 votes)
- This is a really good video that I found:
It explains the formula with simple geometry and algebra, and in animation.(11 votes)
- When using the pi symbol in calculations, the answer comes out differently than if you just use 3.14, which is how the answers on here are calculated. Why not just recommended to use only 3.14 for pi, not the symbol on calculators as shown on the videos? It is frustrating to get sent back to zero when you use your calculator correctly but you should just use one standard calculation of 3.14.(4 votes)
- 3.14 is a simplified version of the real value of pi:
- how can I find volume of a tetrahedron(3 votes)
- A tetrahedron is just a triangular pyramid. If any pyramid has height h and a base of area b, then it's volume is bh/3.(4 votes)
- This article shows how to find the volume of a four sided pyramid, but how do you find the volume of a three sided pyramid.
- What would be the equation to find the volume for a trapezoid prism?(0 votes)
- Any prism volume is V = BH where B is area of base and H is height of prism, so find area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.(7 votes)
- What would be the difference between a triangular prism and a pyramid?(1 vote)
- A prism has the base that goes through the whole figure, so a triangular prism will have two congruent bases that are a triangle. A triangular pyramid has a triangular base and each base vertex that has an edge that goes to a common vertex.(2 votes)
- I encountered a problem where I'm only given the areas of 3 faces, and I need to find the volume. How can I solve this?(1 vote)
- I assume you mean the areas of three faces of a rectangular prism (or some figure whose volume is length x width x height, a parallelepiped), and that no two of your given sides are opposite one another, so you don't have redundant information.
Then the areas of your sides are lw, wh, and lh (length, width, and height). Multiply these together, and you get l²w²h²=(lwh)²=V².
So to get the volume of your figure, multiply your three areas together and take the square root.(2 votes)
- I still don't understand how to find the volume of a pyramid.(1 vote)
- how do you do the volume of a triangular prism that has one of the triangles facing down(2 votes)
- Do not know what you mean, the orientation of the prism does not change the volume or the formula.(0 votes)